Publ. RIMS, Kyoto Univ.
41 (2005), 793–798
Weak Solution of a Singular Semilinear
Elliptic Equation in a Bounded Domain
By
Robert Dalmasso∗
Abstract
We study the singular semilinear elliptic equation ∆u + f (., u) = 0 in D (Ω),
where Ω ⊂ Rn (n ≥ 1) is a bounded domain of class C 1,1 . f : Ω × (0, ∞) → [0, ∞) is
such that f (., u) ∈ L1 (Ω) for u > 0 and u → f (x, u) is continuous and nonincreasing
for a.e. x in Ω. We assume that there exists a subset Ω ⊂ Ω with positive measure
such that f (x, u) > 0 for x ∈ Ω and u > 0 and that Ω f (x, cd(x, ∂Ω)) dx < ∞ for
all c > 0. Then we show that there exists a unique solution u in W01,1 (Ω) such that
∆u ∈ L1 (Ω), u > 0 a.e. in Ω.
§1.
Introduction
Let Ω be a sufficiently smooth (e.g. of class C 1,1 ) bounded domain in Rn
(n ≥ 1). We consider the singular boundary value problem
D (Ω) ,
(1.1)
∆u + f (., u) = 0 in
(1.2)
u ∈ W01,1 (Ω) , f (., u(.)) ∈ L1 (Ω) ,
where f satisfies the following conditions:
(H1) f : Ω × (0, ∞) → [0, ∞). For all u > 0, x → f (x, u) is in L1 (Ω), and
u → f (x, u) is continuous and nonincreasing for a.e. x in Ω;
(H2) There exists Ω ⊂ Ω with positive measure such that f (x, u) > 0 for
x ∈ Ω and u > 0;
Communicated by H. Okamoto. Received July 12, 2004.
2000 Mathematics Subject Classification(s): 35J25, 35J60.
∗ Laboratoire LMC-IMAG, Equipe EDP, Tour IRMA, BP 53, F-38041 Grenoble Cedex 9,
France.
e-mail: [email protected]
c 2005 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.
794
Robert Dalmasso
(H3) For all c > 0
f (x, cd(x, ∂Ω)) dx < +∞ .
Ω
This kind of singularity has been considered by several authors, particularly the case where
f (x, u) = p(x)u−λ ,
λ > 0 [4, 5, 6, 9, 10, 11, and their references].
Lazer and McKenna [10] for instance established the existence and uniqueness of a positive u ∈ C 2 (Ω) ∩ C(Ω) satisfying
∆u + p(x)u−λ = 0 in Ω ,
(1.3)
(1.4)
u = 0 on ∂Ω ,
when p is Hölder-continuous and strictly positive in Ω.
Del Pino [6] proved that if p is a bounded, nonnegative measurable function
which is positive on a set of positive measure, then (1.3)–(1.4) has a unique
positive weak solution in the sense that u ∈ C 1,α (Ω) ∩ C(Ω) satisfies (1.4),
u > 0 in Ω and
pu−λ ϕ
∇u∇ϕ =
Ω
Ω
for all ϕ ∈ Cc∞ (Ω).
Lair and Shaker [9] considered the case
f (x, u) = p(x)g(u) ,
under the following assumptions:
(A0) p ∈ L2 (Ω) is nontrivial and nonnegative;
(A1) g (s) ≤ 0;
(A2) g(s) > 0 if s > 0;
ε
(A3) 0 g(s) ds < ∞ for some ε > 0.
They established the existence of a unique weak solution in the sense that
u ∈ H01 (Ω) satisfies
(∇u∇v − p(x)g(u)v) dx = 0 ∀ v ∈ H01 (Ω) .
Ω
Notice that, when g(u) = u−λ , conditions (A1) and (A3) imply that 0 ≤
λ < 1.
Weak Solution of a Singular Equation
795
Finally Mâagli and Zribi [11] treated the general case f (x, u). However
their assumptions are different from ours and they lead them to the existence
of a weak solution in C(Ω).
Our purpose is to give a general existence and uniqueness result under
sufficiently weak conditions. We shall prove the following theorem.
Theorem 1.
Let Ω ⊂ Rn (n ≥ 1) be a bounded domain of class C 1,1
and let f : Ω × (0, ∞) → [0, ∞) satisfy (H1)–(H3). Then problem (1.1)–(1.2)
has a unique solution.
§2.
Proof of Theorem 1
1) Uniqueness of the solution. We shall need the following lemma ([7,
Lemma 3]).
Lemma 1.
Let p ∈ C 1 (R, R) ∩ L∞ (R) be a nondecreasing function satisfying p(0) = 0. For u ∈ W01,1 (Ω) such that ∆u ∈ L1 (Ω) we have
∆u.p(u) ≤ 0 .
Let u1 , u2 be two solutions of problem (1.1)–(1.2). Let u = u1 − u2 . By
(H1) we have u∆u ≥ 0 a.e. in Ω. Now let p ∈ C 1 (R) ∩ L∞ (R) be a strictly
increasing function satisfying p(0) = 0. Then p(u)∆u ≥ 0 a.e. in Ω. Using
Lemma 1 we deduce that p(u)∆u = 0 a.e. in Ω and therefore ∆u = 0 a.e. in
Ω. Since u ∈ W01,1 (Ω), this implies that u = 0 a.e. in Ω.
2) Existence of a solution. We first recall the following result ([1, Lemme 2.8]).
Lemma 2.
a.e. in Ω.
Let u ∈ W01,1 (Ω) be such that ∆u ≥ 0 in D (Ω). Then u ≤ 0
In the sequel N denotes the set of positive integers.
Lemma 3.
Let j ∈ N . There exists a unique uj ∈ W01,1 (Ω) such that
f (., uj + 1j ) ∈ L1 (Ω), uj ≥ 0 a.e. in Ω and ∆uj + f (., uj + 1j ) = 0 in D (Ω).
Proof. Define
βj (x, u) = f
1
1
x,
− f x, u +
,
j
j
x ∈ Ω, u ≥ 0.
796
Robert Dalmasso
and
x ∈ Ω, u ≤ 0.
βj (x, u) = 0 ,
Then we have:
- For all u ∈ R, x → βj (x, u) is in L1 (Ω);
- R u → βj (x, u) is continuous and nondecreasing for a.e. x in Ω;
- βj (x, 0) = 0 for a.e. x in Ω.
Since f (., 1j ) ∈ L1 (Ω) Theorem 3 in [7] implies the existence of a unique
uj ∈ W01,1 (Ω) satisfying βj (., uj ) ∈ L1 (Ω) and
1
−∆uj + βj (., uj ) = f .,
in D (Ω) .
j
Since ∆uj ≤ 0 in D (Ω), Lemma 2 implies that uj ≥ 0 a.e. in Ω. Therefore we
have f (., uj + 1j ) ∈ L1 (Ω) and
1
∆uj + f ., uj +
= 0 in D (Ω) ,
j
and the lemma is proved.
Lemma 4.
For every j ∈ N there exists aj > 0 such that
uj (x) ≥ aj d(x, ∂Ω) ,
for a.e. x ∈ Ω .
Proof. For ε > 0 we set Ωε = {x ∈ Ω; d(x, ∂Ω) > ε}. Clearly (H2) implies
that, for every j ∈ N , there exist εj > 0 and Mj > 0 such that the function
f˜j defined by
1
˜
fj (x) = min f x, uj (x) +
, Mj 1Ωεj (x), x ∈ Ω ,
j
satisfies f˜j ≡ 0. Let vj be the solution of the following boundary value problem
∆vj + f˜j = 0 in
vj = 0 on
Ω,
∂Ω .
It is well-known (see [8]) that, for 1 < p < ∞, vj ∈ C 1 (Ω) ∩ W 2,p (Ω). We
have ∆(uj − vj ) ≤ 0 in D (Ω), hence by Lemma 2 uj ≥ vj a.e. in Ω. Now the
boundary point version of the Strong Maximum Principle for weak solutions
([12], Theorem 2) implies that there exists aj > 0 such that vj (x) ≥ aj d(x, ∂Ω)
for x ∈ Ω and the lemma follows.
797
Weak Solution of a Singular Equation
For every j ∈ N we have uj +
Lemma 5.
1
j
≥ uj+1 +
1
j+1
a.e. in Ω.
1
)−(uj + 1j ). Using a variant of Kato’s inequality
Proof. Let u = (uj+1 + j+1
(see [3] Lemma A1 in the Appendix) we deduce that
∆u+ ≥ 0 in D (Ω) .
u+ ∈ W01,1 (Ω) (see [1, Lemme 2.7]). Therefore Lemma 2 implies that u ≤ 0
a.e. in Ω and the lemma is proved.
For every j ∈ N we have uj ≤ uj+1 a.e. in Ω.
Lemma 6.
Proof. Using (H1) and Lemma 5 we get
1
− f ., uj+1 +
∆(uj+1 − uj ) = f ., uj +
j
1
j+1
≤ 0 a.e. in Ω ,
and we conclude with the help of Lemma 2.
Now we define
cj =
f
1
x, uj (x) +
dx ,
j
Ω
Using Lemma 4 and Lemma 6 we can write
cj ≤
f (x, a1 d(x, ∂Ω)) dx ,
j ∈ N .
∀ j ∈ N .
Ω
Therefore
(3.1)
sup cj < ∞ .
j∈N
Now we can prove the existence. By (H1) and Lemma 5 j → f (., uj + 1j )
is nondecreasing. (3.1) and the Beppo Levi theorem for monotonic sequences
imply that there exists g ∈ L1 (Ω) such that
1
f ., uj +
→ g in L1 (Ω) as j → ∞ .
j
We have the following estimate [2, Theorem 8]: for 1 ≤ q < N/(N − 1) there
exists Mq > 0 such that
||uj ||W 1,q (Ω) ≤ Mq ||∆uj ||L1 (Ω)
∀ j ∈ N .
798
Robert Dalmasso
Therefore there exists u ∈ W01,1 (Ω) such that uj → u in W01,1 (Ω). By Lemma 6
and the Fischer-Riesz theorem uj → u a.e. in Ω. Lemma 4 and Lemma 6 imply
that u > 0 a.e. in Ω. Clearly we have g = f (., u) and ∆u + f (., u) = 0 in D (Ω).
The proof is complete.
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